25 April 2008

Fractions are not about pizza

Study Suggests Math Teachers Scrap Balls and Slices, from today's New York Times.

The Times article is about a study reported on in today's issue of Science (Jennifer A. Kaminski, Vladimir M. Sloutsky, Andrew F. Heckler1. The Advantage of Abstract Examples in Learning Math. Science 25 April 2008: Vol. 320. no. 5875, pp. 454 - 455). Researchers taught the idea of the group Z3 to some students who weren't familiar with it; some learned it "abstractly" (the elements of the group were represented as funny-looking symbols) and some learned it "concretely" (by considering the slices in a pizza with three slices, or thirds of a measuring cup, or tennis balls in a three-ball can). It seems that the ones who learned the "abstract" version more easily picked up the rules of yet another "concrete" version (a children's game) than those who learned the original "concrete" version.

The Science authors claim that this is because "Compared with concrete instantiations, generic instantiations present minimal extraneous information and hence represent mathematical concepts in a manner close to the abstract rules themselves." This seems like the whole point of mathematics -- a lot of what we do as mathematicians is to strip away extraneous details of a problem while retaining those that are actually significant. If you learn about fractions by thinking about slices of pizza, perhaps you will always think that fractions are about pizza. And then whenever you hear about them, you'll think "where's lunch"?


misha said...

I am unconvinced. My hunch and personal experience is that the concept is learned the best when it introduced as a tool to solve a nontrivial problem, and it looks like the researchers haven't tried that. I also suspect that the students that think about fractions as pizza or apple slices will be less likely to add fractions by adding their numerators and their denominators, inapproprianely using an intuitive and pervasive concept of a homomorphism, that they even never heard about.

Paul said...

I think this study misses what was traditionally the point of real-world examples--namely teaching students how to apply math to the real world.

I know that students can be drilled on algebra as an abstract game and can learn to solve equations very fast. It may also be true that they will learn it faster than if their brains were weighted down with thoughts of passing trains, apples, or pizzas. If that's all the study showed, it seems unsurprising.

But if they ever plan to apply this math, it won't help them very much unless they see it mapped onto reality.

I may be biased, because I have always intuitively grounded my mathematical thoughts in concrete examples. I am not the fastest equation solver and I make mistakes. When I set up the train example, for instance, I wrote "40t+50(t-1)=400" which is correct and then tried to skip a step, wrote 90t=350, got the wrong answer as I discovered when plugging it back it, found my mistake and saw it should be 90t=450 (my t is the time from 6pm rather than 7pm as in the article).

I have a PhD in a field that required me to write many mathematical proofs over the years. I know many people who would laugh at my algebra mistakes, but I also know many people who have a lot more trouble mapping their abstraction onto the real world. You don't even need to do that if you're a pure mathematician, but engineers need to do it.

There are already excellent symbolic math packages out there, so I tend to side (perhaps out of bias) with those who emphasize the ability to map the world onto mathematics over the ability to crunch through the formalism (of course it's better if you can do both).

Without the formalism, you will not get very drawing pizza slices and apples. That's why these powerful notational tools were invented. We want students to learn that formalism. For the ones who will pick it up as an abstract game, that's great for them. Others who don't find it such a fun game may need some persuasion that it has some purpose beyond busywork.

Finally, it would be useful to know the ages of students in this study. By the time students have had a few years of grade school, they should be able to think abstractly. At that point, it really is a bit of a crutch to be making everything concrete. I just think that it would be terrible if students were to learn a pure formalism without any attempt to map it onto reality. What on earth is the purpose of learning algebra merely as a series of arbitrary rewriting rules?

Anonymous said...

I don't have access to the Science article, and I imagine there was more than one topic being tested, but as far as Z3 goes... slices and balls and thirds of a measuring cup are really not the proper metaphors to be using. Of course kids taught that kind of nonsense aren't going to do as well!

Beyond that, I think it's important to learn abstract topics using multiple concrete examples, so that one's brain can grasp the common structure between them. Of course, what counts as 'concrete' can be a matter of taste. Integers seem pretty concrete to me.

Paul said...

One other comment about extraneous details. The reason word problems contain extraneous details is not because they help solve the problem, but because they present an extra challenge of abstract thinking.

Obviously, if you write 3x+x=8, this is easier to solve than if you write "There is a pizza cut into eight slices. Mary can eat three slices as fast as John can eat one. They start eating the pizza at the same time. How many slices does John eat?"

The names are extraneous. The process of eating is extraneous. The fact that it is pizza is extraneous. None of this will help you learn how to do algebra. On the other hand, being spoon fed the equation won't help you to learn how to throw away extraneous details and model a situation in abstract terms.

I am very much against the idea of focusing students on formalism primarily. It sounds suspiciously like just another way to put the minimal effort into getting them to score well on standardized tests without asking what is the value of these tests.

Anonymous said...

Paul said:
but I also know many people who have a lot more trouble mapping their abstraction onto the real world. You don't even need to do that if you're a pure mathematician, but engineers need to do it...


The reason word problems contain extraneous details is not because they help solve the problem, but because they present an extra challenge of abstract thinking.

I couldn't agree more. The challenge is to take something concrete, deduce the meaningful abstractions, reason about them, then map your conclusions back to the concrete. Arguing that you [not you in particular, Iz] only need half the steps says to me, "I don't really understand the whole problem," not, "It's just this easy."

I've probably mentioned this before, but I reiterated because it gets me all riled up. My niece is taking trigonometry. In the course of a couple of classes they went from:
Graph 5sin(12x + 3π/8) + 6
to something like:
You are sitting in a car on a ferris wheel. When Joebob gets on two cars behind you, you are X feet above the ground, 38 seconds later, you are at the bottom of the ferris wheel. What is the formula that describes how high you are off the ground.

There wasn't much meaningful teaching going on in between.

Paul said...

Actually, I agree that abstract concepts need to be understood with generic symbols in order to be used most effectively. If I were to explain the Chinese Remainder Theorem, I'm better off using mod notation than imagining a set of spinning clock dials (though as I think about it, I still feel that this could be a useful intuition building exercise to reinforce the formalism). On the other hand, if I had to explain modular arithmetic to a kindergarten student, a clock (not pizza or apples) seems a logical starting point.

I am not sure what I said that suggests I believe that "abstract concepts can be hard to transfer to other situations." Once you've mastered an abstract concept in abstract terms, you can apply it. It's not always trivial; I can think of physics exams from years ago where the hard part was figuring out how to set up the appropriate formula rather than how to solve it.

I also agree that some kind of model based on pizza slices is inherently more difficult to transfer to other situations than a model with extraneous details removed. The first step would be to remove it from context before applying it to a new context.

The results of the study seem reasonable, and some attempts to make learning concrete are probably misguided (particularly if applied beyond the early stages of math education).

But my kneejerk opposition is to the potential conclusion that all we ever need to teach is formalism. I'm not buying it. I have seen people who can manipulate digits but don't understand number in the abstract. I have seen people who know every formula in their Schaum's outline but could not solve a physics problem unless you pointed out the right one and showed them which numbers go where. If we only teach formalism we will get kids who master the formalism. This will not make them into intuitive mathematicians.

Paul said...

I may look at Zane Kaminski's links if I get a chance. I started looking at the first one.

The conclusions seem reasonable, and some of my reaction may be connected to how it was presented in the NYT article.

The problem of treating inconsequential details as important rings true. I was recently trying to discuss numbers with my four year old based on an exercise in his preschool in which he made "bundles" of ten sticks--"and a rubber band!" That rubber band was really an obstacle, but was probably of far greater interest to him than a bunch of sticks. Maybe it is better if he just learns decimal notation by rote. But I would prefer that he can move between the concrete and the abstract and I was hoping to reinforce an existing lesson.

It seems sort of obvious to me that a good abstraction is crucial to progress. Compare Roman numerals to Arabic numerals for instance, or modern calculus notation with early attempts to use infinitesimals. When there are highly optimized notational systems, you don't want to waste time making ad hoc ones any more than you'd invent a new alphabet.

If the question is just where to start, at the abstract or the concrete, then maybe the studies are correct that it is more effective to start abstract and map to the concrete. Personally, I don't think there is such a huge hazard that explaining a problem in terms of pizza will send students into Homer-Simpson-like reveries, but I guess it could happen.

The main problem is if you propose to teach the abstraction and stop there. I simply don't believe that everyone naturally figures out how to map their abstract system onto applicable reality. In fact a large part of peer-reviewed research is nothing more than a scientist finding that a set of observations matches a pre-existing model in a surprising way. If the mapping from abstraction to reality was always so easy, there would never be any such surprise.

Anonymous said...

Zane, thanks for those links. Interesting stuff, but as far as the particular study with teaching Z3 to the kids (the 3rd link), I have to disagree strongly with the conclusion, based on my own attempt at decyphering the second task. The problem is that the 'concrete' test subjects are being trained on Z3, and then being challenged with C3. The 'abstract' test subjects get trained and challenged with C3.

Before anyone points out the obvious fact that those are the same group, the problem is that Z3 has an interpretation in terms of arithmetic that is going to interfere strongly with seeing the pattern in C3 that does not have an arithmetic interpretation. Now, you could argue that in this case arithmetic itself is an example of concreteness that gets in the way of understanding abstract groups, and I would agree with you. But that was not what was being implied by the description of the study.

Anonymous said...

I just read the NYT article.

What's interesting is that it goes:
Test group - learn this abstract stuff
Control group - Solve these word problems

Both groups using what you've just learned solve this new problem.

Isn't that the hardest part of building a math model of something, figuring out which math applies to it?

Ben Allen said...

As a former math teacher, I believe that using either abstraction or concrete examples without the other is madness. The whole point is to be able to convert between two.

I'm not surprised the students who learned via abstract methods were better able to transfer the concepts, because that's the point of abstract reasoning. But concrete methods have an equally valid point: to give motivation for the abstractions and to show how they can be applied.